For the intake process, here are some steps
1. Identify the scope and which functional and technical area it falls in.
2. Bring the stakeholders into one meeting(typically- tech leads, managers, SMEs)
3. Yes, based on T-shirt sizes, estimate the budget.
4. Fill the necessary documentation based on the size of the project(different approvals from sponsors, finance etc)
one example: less than 500K, internal scope budget, greater than 500K, Director and Sponsors formal approval etc.
Hope this helps.